Related papers: Evaluating a double integral using Euler's method …
In this study, we employ Euler's method and Richardson's extrapolation to solve a triple integral, which is then transformed into a third-order initial value problem. Our objective is to resolve the computational challenges associated with…
We describe an algorithm, based on Euler's method, for solving Volterra integro-differential equations. The algorithm approximates the relevant integral by means of the composite Trapezium Rule, using the discrete nodes of the independent…
Richardson extrapolation is a classical technique from numerical analysis that can improve the approximation error of an estimation method by combining linearly several estimates obtained from different values of one of its hyperparameters,…
We propose an extrapolation technique that allows accuracy improvement of the discrete dipole approximation computations. The performance of this technique was studied empirically based on extensive simulations for 5 test cases using many…
The cubic spline interpolation method, the Runge--Kutta method, and the Newton-Raphson method are extended to dual versions (developed in the context of dual numbers). This extension allows the calculation of the derivatives of complicated…
In this paper we investigate the use of Richardson extrapolation to estimate the convergence rates for numerical solutions to advection problems involving discontinuities. We use modified equation analysis to describe the expectation of the…
We propose a multi-step Richardson-Romberg extrapolation method for the computation of expectations $E f(X_{_T})$ of a diffusion $(X_t)_{t\in [0,T]}$ when the weak time discretization error induced by the Euler scheme admits an expansion at…
Using elementary methods, we define and derive a particular weighted average of the trapezoidal and composite trapezoidal rules and show that this approximation, as well as its composite, is straightforward in computation. This…
We present a quantum algorithm to solve systems of linear equations of the form $A\mathbf{x}=\mathbf{b}$, where $A$ is a tridiagonal Toeplitz matrix and $\mathbf{b}$ results from discretizing an analytic function, with a circuit complexity…
We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on…
We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the…
In this paper we propose two different primal-dual splitting algorithms for solving inclusions involving mixtures of composite and parallel-sum type monotone operators which rely on an inexact Douglas-Rachford splitting method, however…
We introduce a technique to estimate a linear operator by embedding it in a family $A_t$ of operators, $t\in(\sigma_0,\infty)$, with suitable curvature properties. One can then estimate the norm of each $A_t$ by bounds that hold in the…
For over a century, extrapolation methods have provided a powerful tool to improve the convergence order of a numerical method. However, these tools are not well-suited to modern computer codes, where multiple continua are discretised and…
We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the…
We propose a primal--dual technique that applies to infinite dimensional equality constrained problems, in particular those arising from optimal control. As an application of our general framework, we solve a control-constrained double…
In this brief, we discuss the implementation of a third order semi-implicit differentiator as a complement of the recent work by the author that proposes an interconnected semi-implicit Euler double differentiators algorithm through Taylor…
A class of linear parabolic equations are considered. We derive a common framework for the a posteriori error analysis of certain second-order time discretisations combined with finite element discretisations in space. In particular we…
In this paper, we introduce the cubature formula for Stochastic Volterra Integral Equations. We first derive the stochastic Taylor expansion in this setting, by utilizing a functional It\^{o} formula, and provide its tail estimates. We then…
This paper describes applications of extrapolation for the computation of coefficients in an expansion of infrared divergent integrals. An extrapolation procedure is performed with respect to a parameter introduced by dimensional…